Thermodynamics of light management in photovoltaic devices

This paper elaborates a comprehensive theory of the thermodynamics of light management in solar cells explicitly considering imperfect light trapping, parasitic absorption and nonradiative recombination losses. A quantitative description of the entropic losses that reduce the open-circuit voltage and the energy conversion efficiency from the radiative limit towards realistic situations is presented. The theory embraces the fundamental limits for idealized solar cell devices given by the Yablonovitch limit and the Shockley-Queisser limit. We discriminate between reversible and irreversible entropic loss processes for four fundamental light management concepts: (i) conventional light trapping as an integral part of the device, (ii) geometric concentration of incident light, (iii) angular restriction of incoming and outgoing light, and (iv) light concentration by luminescent solar collectors. Based on this discrimination, a comprehensive discussion of the interplay between the loss processes and light management is presented. As part of this analysis, a new figure of merit for efficient light trapping in solar cells is introduced as well as an example of a deterministic light trapping concept which induces almost optimal light trapping.

Figure 1

Schematic illustrations of four fundamental light-management concepts: (a) conventional light trapping as an integral part of the device, (b) geometric concentration of incident light, (c) angular restriction of incoming and outgoing light, and (d) luminescent solar collectors. The acceptance angle (${\theta }_{\mathrm{in}}$) and emission angle (${\theta }_{\mathrm{out}}$) of the light cones of incident and emitted light are shown.

Figure 2

Thermodynamic limitations of the open-circuit voltage $V_{\mathrm{OC}}$ of a single solar cell (idealized semiconductor with $E_g$ = 1.38 eV, step-function-like absorptance and $Q_{\mathrm{e}}^{\mathrm{LED}}\left(E\right)={10}^{-3}$) limiting also the overall power conversion efficiency since $\eta \le J_{\mathrm{SC}}{V}_{\mathrm{OC}}/P_{\mathrm{in}}$.

Figure 3

(a) The enhancement of the open-circuit voltage $V_{\mathrm{OC}}$ relative to the open-circuit voltage in the radiative limit $V_{\mathrm{OC}}^{\mathrm{rad},1}$ is shown as a function of the internal luminescence quantum efficiency $Q_{\mathrm{i}}^{\mathrm{lum}}$ for the four fundamental light-management concepts: conventional light trapping ($c_{\mathrm{geo}}=c_{\mathrm{ang}}=1$), geometric concentration of incident light ($c_{\mathrm{geo}}>1,c_{\mathrm{ang}}=1$), angular restriction of incoming and outgoing light ($c_{\mathrm{geo}}=1,c_{\mathrm{ang}}>1$), and luminescent solar collectors (fc $c_{\mathrm{geo}}>1$). (b) The change in open-circuit voltage losses $\Delta V_{\mathrm{OC}}$ caused by parasitic light absorption (solid: $p_a=0$; dashed: $p_a=0.1$) as a function of the internal luminescence quantum efficiency $Q_{\mathrm{i}}^{\mathrm{lum}}$ is shown for geometric concentrators ($p_e=0.5$; $c_{\mathrm{geo}}=1,c_{\mathrm{ang}}=1$), angular restriction ($p_e=0.005$; $c_{\mathrm{geo}}=1,c_{\mathrm{ang}}=100$), and the fluorescent concentrator ($p_e=0.05$; fc $c_{\mathrm{geo}}=10$). The dashed lines correspond to the influence of parasitic light absorption with $p_a=0.1$.

Figure 4

(a) Schematic illustration of the light scattering and light propagation for the partial Lambertian light-trapping scheme. (b) Schematic illustration of light propagation for the exemplary deterministic light-trapping scheme discussed in this contribution. The light cone is preserved during the 32 passes through the solar cell. The mapping rules for the redirection of the light at the front and rear interface are shown in (c).

Figure 5

(a) The absorptance $A$ of an ideal solar cell employing the partial Lambertian light-trapping scheme as a function of the thickness-corrected absorption coefficient ($\alpha w$) with $w$ being the thickness of the solar cell and $\alpha$ the energy dependent absorption coefficient. The probability $p_{\lambda }$, which denotes probability of incoming light being scattered at the front surface into Lambertian distribution, is varied from no light trapping ($p_{\lambda }=0$) to Lambertian light trapping ($p_{\lambda }=1$). (b) The path length distribution $P$ as a function of the relative path length ($l/w$).

Figure 6

(a) The absorptance $A$ of an ideal solar cell employing the deterministic light trapping, the partial Lambertian light-trapping scheme, as well as the optimal light trapping as an upper boundary are plotted as a function of the thickness-corrected absorption coefficient ($\alpha w$) with $w$ being the thickness of the solar cell and $\alpha$ the energy dependent absorption coefficient. (b) The path length distribution $P$ is shown for the Lambertian light-trapping scheme and the deterministic light-trapping scheme as a function of the relative path length ($l/w$).

Figure 7

Effective path lengths $〈l〉$ and ${〈1/l〉}^{-1}$ for partial Lambertian light trapping (open symbols), the deterministic light-trapping scheme (full symbols), and the value 4$n^2$ (dashed line), which is the maximum value for $〈l〉$ and an upper bound for ${〈1/l〉}^{-1}$.

Figure 8

Short-circuit current density $J_{\mathrm{SC}}$, open-circuit voltage $V_{\mathrm{OC}}$, and energy conversion efficiency $\eta$ of a solar cell applying the deterministic light-trapping scheme as a function of the effective thickness (${\alpha }_{0}w$). (a)–(c) The parasitic absorption is neglected $A_{\mathrm{BR}}$ = 0, and the internal luminescence quantum efficiency $Q_{\mathrm{i}}^{\mathrm{lum}}$ is varied from 0.9 to 9 10${}^{-11}$. (d)–(f) $A_{\mathrm{BR}}$ is varied from 10${}^{-3}$ to 10${}^0$, and the $Q_{\mathrm{i}}^{\mathrm{lum}}$ is set to 0.5. For comparison, the reference quantities in the radiative limit ($J_{\mathrm{SC}}^{\max }$, $V_{\mathrm{OC}}^{\mathrm{rad}}$, ${\eta }_{\mathrm{rad}}$) for zero parasitic absorptance and an infinitesimal thick solar cell are shown.

Figure 9

The maximum energy conversion efficiency ${\eta }_{max}$ for the deterministic light-trapping scheme (solid symbols) and the Lambertian light-trapping scheme (open symbols) as (a) a function of internal luminescence quantum efficiency $Q_{\mathrm{i}}^{\mathrm{lum}}$ and (b) as a function of the parasitic absorption at the back reflector $A_{\mathrm{BR}}$. For comparison, the maximum energy conversion efficiency radiative limit ${\eta }_{\mathrm{rad}}^{\max }$ is shown.

Figure 10

The open-circuit voltage losses $\Delta V_{\mathrm{OC}}^{\mathrm{r}}$ due to nonradiative recombination as a function of internal luminescence quantum efficiency $Q_{\mathrm{i}}^{\mathrm{lum}}$ are calculated by term 4 in Eq. (26) (solid lines) and compared to the Monte Carlo simulations of deterministic light trapping (open symbols) for (a) a fixed $p_a$ = 0 and $p_e$ = 0.005, 0.05, and 0.5 and (b) a fixed $p_e$ = 0.05 and $p_a$ = 0, 0.25, and 0.75.

Figure 11

(a) Open-circuit voltage $V_{\mathrm{OC}}$ of solar cells employing the deterministic light trapping ($c_{\mathrm{ang}}$ = $c_{\mathrm{geo}}$ = 1), deterministic light trapping with geometrical concentration ($c_{\mathrm{geo}}$ = 10) and angular concentration ($c_{\mathrm{ang}}$ = 100) are shown as a function of the internal luminescence quantum efficiency $Q_{\mathrm{i}}^{\mathrm{lum}}$. The parasitic absorption at the back reflector $A_{\mathrm{BR}}$ is 2%. The discrimination of the four fundamental entropic loss processes decreasing the $V_{\mathrm{OC}}$ as described in Eq. (26) is shown for the three light-trapping schemes in (b)–(d), respectively.